PROBABILITY GUIDE TO GAMBLING

PROBABILITY GUIDE TO

GAMBLING

The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets

Ctlin Brboianu

INFAROM Publishing Applied Mathematics office@



ISBN-10 9738752027 ISBN-13 9789738752023

Publisher: INFAROM Author and translator: Ctlin Brboianu

Correction Editor: Sara M. Stohl

Copyright ? INFAROM 2009

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of tables, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks.

Duplication of this publication or parts thereof is permitted only under the provisions of Copyright Laws and permission for use must always be obtained from INFAROM.

CONTENTS (of the complete edition)

Introduction ................................................................ 5 Probability Theory Basics ................................................ 15

Fundamental notions ...................................................... 16 Sets ........................................................................ 16 Functions ................................................................. 18 Boole algebras ........................................................... 20 Sequences of real numbers. Limit .................................... 22 Series of real numbers .................................................. 28

Measure theory basics .................................................... 29 Sequences of sets ........................................................ 30 Tribes. Borel sets. Measurable space ................................. 31 Measure .................................................................. 34

Field of events. Probability .............................................. 37 Field of events ........................................................... 39 Probability on a finite field of events ................................. 45 Properties of probability ................................................ 48 Probability -field ...................................................... 54 Independent events. Conditional probability ........................ 56 Total probability formula. Bayes's theorem ...................... 58 Law of large numbers ................................................... 59 Discrete random variables ............................................. 61 Distribution function ................................................. 64 Classical discrete probability distributions ........................ 65 Bernoulli scheme ................................................... 65 Poisson scheme ..................................................... 67 Polynomial scheme ................................................ 69 Scheme of nonreturned ball ....................................... 70 Convergence of sequences of random variables .................. 71 Law of large numbers .............................................. 72

Combinatorics .............................................................. 75 Permutations ............................................................. 75 Arrangements ............................................................ 76 Combinations ............................................................ 77

The Mathematics of Games of Chance ................................. 79 Experiments, events, probability fields .............................. 80 Probability calculus ..................................................... 88 Properties of probability ............................................. 88 Odds and probability ................................................. 91 Combinations ......................................................... 92 Expectation .............................................................. 93 Relative frequency. Law of Large Numbers ........................ 96

Relativity of probability results ....................................... 100 The Guide of Numerical Results ........................................ 105

Dice ......................................................................... 106 Two dice .................................................................. 106 Three dice ................................................................ 109

Slots ......................................................................... 111 Three reels ............................................................... 111 Four reels ................................................................. 112

Roulette .................................................................... 118 Simple bets ............................................................... 121 Complex bets ............................................................ 124 Equivalent bets .......................................................... 127 Betting on a colour and on numbers of the opposite colour ...... 129 The martingale .......................................................... 147

Baccarat ................................................................... 151 Blackjack .................................................................. 181 Classical Poker ........................................................... 201

Initial probabilities on the first card distribution for your own hand ........................................................................... 205

Prediction probabilities after the first card distribution and before the second for your own hand ..................................... 214

Prediction probabilities for opponents' hands ....................... 239 Odds of holding three of a kind or better .......................... 241

Texas Hold'em Poker ................................................... 248 Immediate odds ......................................................... 252 Preflop odds ........................................................... 252 Flop odds ............................................................... 262 Turn odds .............................................................. 274 Other odds ............................................................. 275 Opponents' hands probabilities ....................................... 279

Lottery ..................................................................... 280 General formula of the winning probability ......................... 283 Cumulated winning probabilities ..................................... 297 Enhancing the winning probability ................................... 299

Sport Bets .................................................................. 304 The Probability-based Strategy ......................................... 311

References ................................................................. 329

Introduction

Probability theory is a formal theory of mathematics like many others, but none of them raised so many questions about its interpretations and applicability in daily life as this theory does. Even though many of these questions have found no satisfactory answer yet, probability still remains the only theory that models hazards through mathematical methods, even if it operates on a minute part of what a hazard would mean.

Owing to its psychological impact on human concerns in daily life, probability theory has gained considerable popularity among ordinary people, regardless of whether they have a mathematical background. Moreover, people refer to probability and statistics anytime they need additional information about the occurrence of an event.

Laymen and scientists alike are fascinated by probability theory because it has multiple models in nature, it is a calculus tool for other sciences and the probability concept has major philosophical implications as well.

But no matter how paradoxically it may look, the popularity of probability theory is not that beneficial for average people. The relativities of the term probability, even if related only to the mathematical definition, may introduce a lot of errors into the qualitative and quantitative interpretation of probability, especially as a degree of belief.

These interpretation errors, as well as that false certainty psychologically introduced by the numerical result of measuring an event, turn probability calculus into a somewhat dangerous tool in the hands of persons having little or no elementary mathematical background. This affirmation is not at all hazardous, because probabilities are frequently the basis of decisions in everyday life. Due to a natural need that is more or less rigorously justified, humans consistently refer to statistics; therefore, probability has become a real decision-making tool.

Here is an example--otherwise unwanted--of making a decision based on probability is the following:

Your doctor communicates the stages of evolution of your disease: if you won't have an operation, you have a 70 percent chance of living, and if you'll have the operation, you have a 90 percent chance of a cure, but there is a 20 percent chance that you will die during the operation.

Thus, you are in a moment when you have to make a decision, based on personal criteria and also on communicated figures (their estimation was performed by the doctor according to statistics).

In most cases of probability-based decisions, the person involved performs the estimation or calculus. Here is a simple example:

You are in a phone booth and you must urgently communicate important information to one of your neighbors (let us say you left your front door open). You have only one coin, so you can make only one call. You have two neighboring houses. Two persons live in one of them and three persons live in the other. Both their telephones have answering machines. Which one of the two numbers will you call?

The risk is that nobody will be at the home you call and the coin will be lost when the answering machine starts. You could make an aleatory choice, but you could also make the following decision: "Because the chances for somebody to be home are bigger in the case of house with three persons, I will call there."

Thus, you have made a decision based on your own comparison of probabilities. Of course, the only information taken into account was the number of persons living in each house.

If other additional information--such the daily schedules of your neighbors--is factored in, the probability result might be different and, implicitly, you might make a different decision.

In the previous example, the estimation can be made by anyone because it is a matter of simple counting and comparison. But in most situations, a minimum knowledge of combinatorics and the calculus of probabilities are required for a correct estimation or comparison.

Millions of people take a chance in the lottery, but probably about 10 percent of them know what the winning probabilities really are. Let us take, for example, the 6 from 49 system (six numbers are drawn from 49 and one simple variant to play has six numbers; you win with a variant having a minimum of four winning numbers).

The probability of five numbers from your variant being drawn is about 1/53992, and the probability of all six numbers being drawn (the big hit!) is 1/13983816.

For someone having no idea of combinations, these figures are quite unbelievable because that person initially faces the small numbers 5, 6 and 49, and does not see how those huge numbers are obtained. In fact, this is the psychological element that a lottery company depends on for its system to work.

If a player knew those figures in advance, would he or she still play? Would he or she play less often or with fewer variants? Or would he or she play more variants in order to better the chances? Whatever the answers to these questions may be, those probabilities will influence the player's decision.

There are situations where a probability-based decision must be made--if wanted--in a relatively short time; these situations do not allow for thorough calculus even for a person with a mathematical background.

In gambling, such decisional situations are encountered all over: you ask yourself which combination of cards it is better to keep and which to replace in a five draw poker, if raise or not after flop in Hold'em, if ask for an additional card in blackjack when you have good points, etc.

If probability is used as a criterion in making gaming decisions, you have to know in advance probabilities of the events related to your own play, as well as probabilities related to opponents' play and compare them.

We estimate, approximate, communicate and compare probabilities daily, sometimes without realizing it, especially to make favorable decisions. The methods through which we perform these operations could not be rigorous or could even be incorrect, but the need to use probability as criterion in making decisions generally has a precedence. In addition, establishing a certain threshold from which the chance of an event occurring attributes to it the quality of being probable or very probable is a subjective choice.

Psychologically speaking, the tendency of novices to grant the word probability a certain importance is generally excessive in two ways: the word is granted too much importance--the figures come to represent the subjective absolute degree of trust in an event

occurring--or too little importance--so many times, an equal sign is put between probable and possible and the information provided by numbers is not taken into account.

This could be explained by the fact that human beings automatically refer to statistics in any specific situation, and statistics and probability theory are related. We usually take a certain action as result of a decision because statistically, that action led to a favorable result in a number of previous cases. In other words, the probability of getting a favorable result after that action is acceptable. This decisional behavior belongs to a certain human psychology and the human action is generally not conditioned by additional knowledge.

Although statistics and even probability do not provide any precise information on the result of a respective action, the decision is made intuitively, without reliance on scientific proof that the decision is optimum.

What is, in fact, the motivation behind such general behavior of appealing to statistics? Is there a rational motivation or it is only a matter of human biological structures? The answer is somewhere at the middle and can be explained in large part by psychology.

All along, man felt safe as an individual only when he was grounded on something sure and perceptible. The human mind also submits to this principle. Practical statistics is a collection of sure results; namely, frequencies of events that already happened, and these happenings are a certainty. Unlike statistics, the prediction, by estimating a degree of belief, refers to events that have not happened yet and their occurrence is an uncertainty, so the human mind classifies them in the category of unsafe and tends to increase somehow their sureness by transferring sure things (the statistical results) upon them.

Although humans act in the real word in uncertain conditions, in an unsure environment, the human mind perceives this thing as an anomaly and tries to ameliorate it by elaborating degrees of belief that come from sure environments, such as practical statistics.

This psychic process of unconditioned migration to certainty environments is a reflex action of possibly ancestral origin and may be changed only through a profound study of notions of frequency, probability and degree of belief.

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